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Theorem divsfval 15531
Description: Value of the function in qusval 15526. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 12-Aug-2015.)
Hypotheses
Ref Expression
ercpbl.r  |-  ( ph  ->  .~  Er  V )
ercpbl.v  |-  ( ph  ->  V  e.  _V )
ercpbl.f  |-  F  =  ( x  e.  V  |->  [ x ]  .~  )
Assertion
Ref Expression
divsfval  |-  ( ph  ->  ( F `  A
)  =  [ A ]  .~  )
Distinct variable groups:    x,  .~    x, A    x, V    ph, x
Allowed substitution hint:    F( x)

Proof of Theorem divsfval
StepHypRef Expression
1 ercpbl.v . . . . 5  |-  ( ph  ->  V  e.  _V )
2 ercpbl.r . . . . . 6  |-  ( ph  ->  .~  Er  V )
32ecss 7423 . . . . 5  |-  ( ph  ->  [ A ]  .~  C_  V )
41, 3ssexd 4543 . . . 4  |-  ( ph  ->  [ A ]  .~  e.  _V )
5 eceq1 7417 . . . . 5  |-  ( x  =  A  ->  [ x ]  .~  =  [ A ]  .~  )
6 ercpbl.f . . . . 5  |-  F  =  ( x  e.  V  |->  [ x ]  .~  )
75, 6fvmptg 5961 . . . 4  |-  ( ( A  e.  V  /\  [ A ]  .~  e.  _V )  ->  ( F `
 A )  =  [ A ]  .~  )
84, 7sylan2 482 . . 3  |-  ( ( A  e.  V  /\  ph )  ->  ( F `  A )  =  [ A ]  .~  )
98expcom 442 . 2  |-  ( ph  ->  ( A  e.  V  ->  ( F `  A
)  =  [ A ]  .~  ) )
106dmeqi 5041 . . . . . . . 8  |-  dom  F  =  dom  ( x  e.  V  |->  [ x ]  .~  )
112ecss 7423 . . . . . . . . . . 11  |-  ( ph  ->  [ x ]  .~  C_  V )
121, 11ssexd 4543 . . . . . . . . . 10  |-  ( ph  ->  [ x ]  .~  e.  _V )
1312ralrimivw 2810 . . . . . . . . 9  |-  ( ph  ->  A. x  e.  V  [ x ]  .~  e.  _V )
14 dmmptg 5339 . . . . . . . . 9  |-  ( A. x  e.  V  [
x ]  .~  e.  _V  ->  dom  ( x  e.  V  |->  [ x ]  .~  )  =  V )
1513, 14syl 17 . . . . . . . 8  |-  ( ph  ->  dom  ( x  e.  V  |->  [ x ]  .~  )  =  V
)
1610, 15syl5eq 2517 . . . . . . 7  |-  ( ph  ->  dom  F  =  V )
1716eleq2d 2534 . . . . . 6  |-  ( ph  ->  ( A  e.  dom  F  <-> 
A  e.  V ) )
1817notbid 301 . . . . 5  |-  ( ph  ->  ( -.  A  e. 
dom  F  <->  -.  A  e.  V ) )
19 ndmfv 5903 . . . . 5  |-  ( -.  A  e.  dom  F  ->  ( F `  A
)  =  (/) )
2018, 19syl6bir 237 . . . 4  |-  ( ph  ->  ( -.  A  e.  V  ->  ( F `  A )  =  (/) ) )
21 ecdmn0 7424 . . . . . 6  |-  ( A  e.  dom  .~  <->  [ A ]  .~  =/=  (/) )
22 erdm 7391 . . . . . . . . 9  |-  (  .~  Er  V  ->  dom  .~  =  V )
232, 22syl 17 . . . . . . . 8  |-  ( ph  ->  dom  .~  =  V )
2423eleq2d 2534 . . . . . . 7  |-  ( ph  ->  ( A  e.  dom  .~  <->  A  e.  V ) )
2524biimpd 212 . . . . . 6  |-  ( ph  ->  ( A  e.  dom  .~ 
->  A  e.  V
) )
2621, 25syl5bir 226 . . . . 5  |-  ( ph  ->  ( [ A ]  .~  =/=  (/)  ->  A  e.  V ) )
2726necon1bd 2661 . . . 4  |-  ( ph  ->  ( -.  A  e.  V  ->  [ A ]  .~  =  (/) ) )
2820, 27jcad 542 . . 3  |-  ( ph  ->  ( -.  A  e.  V  ->  ( ( F `  A )  =  (/)  /\  [ A ]  .~  =  (/) ) ) )
29 eqtr3 2492 . . 3  |-  ( ( ( F `  A
)  =  (/)  /\  [ A ]  .~  =  (/) )  ->  ( F `  A )  =  [ A ]  .~  )
3028, 29syl6 33 . 2  |-  ( ph  ->  ( -.  A  e.  V  ->  ( F `  A )  =  [ A ]  .~  )
)
319, 30pm2.61d 163 1  |-  ( ph  ->  ( F `  A
)  =  [ A ]  .~  )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 376    = wceq 1452    e. wcel 1904    =/= wne 2641   A.wral 2756   _Vcvv 3031   (/)c0 3722    |-> cmpt 4454   dom cdm 4839   ` cfv 5589    Er wer 7378   [cec 7379
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fv 5597  df-er 7381  df-ec 7383
This theorem is referenced by:  ercpbllem  15532  qusaddvallem  15535  qusgrp2  16882  frgpmhm  17493  frgpup3lem  17505  qusring2  17926  qusrhm  18538
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